# Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels

arXiv: https://arxiv.org/abs/2211.01804 · HTML: https://mlsanity.com/paper/2211.01804/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2022.

## Built on by (most cited first, 16 of 16)

- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2301.11624/) (2023)
- [Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms](https://mlsanity.com/paper/2401.08260/) (2024)
- [Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions](https://mlsanity.com/paper/2408.07498/) (2024)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Ultra-fast feature learning for the training of two-layer neural networks in the two-timescale regime](https://mlsanity.com/paper/2504.18208/) (2025)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows](https://mlsanity.com/paper/2504.07820/) (2025)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Wasserstein gradient flows for Coulomb discrepancies](https://mlsanity.com/paper/2607.12579/) (2026)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For λ-convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not λ-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.